> On 10/14/2013 4:01 AM, Peter Percival wrote:>> Hetware wrote:>>>>>>>> The statement "Let f(t) be a continuous function for all real numbers t">>> has a concise meaning.>>>> There's nothing wrong with that (one can easily prove that there are>> such functions). What is verboten is defining f with some expression in>> t and _then_ assuming it is continuous.>>>> The more I think about this, the more ambiguous it seems.>> One way to prove something in mathematics is to propose the opposite,> and show that the proposition leads to a contradiction. So let's try> that with the definition f(t) = (t^2-9)/(t-3) without giving> continuity as part of the definition.>> Now, for the purpose of finding a contradiction, let us propose that> f(t) is globally continuous. What does that entail? f(t) has a> definite finite value for every real number t, and that value is f(t)> = limit[f(p), p->t].>> Now, I concede that (t^2-9)/(t-3) = 0/0 when t = 3, and 0/0 is> meaningless. It is not, however, a contradiction.

That seems to me as much of an explicit contradiction as you could want.It may seem odd because the contradiction comes immediately from thedefinition of f rather than after some few lines of reasoning, but it'sa contradiction none the less: assume f continuous everywhere, oops fnot even defined at 3. How is that not a contradiction?